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RbEu(Fe1−xNi

x)4As4: From a ferromagnetic superconductor to a superconducting

ferromagnet

Yi Liu,1, ∗ Ya-Bin Liu,1, ∗ Ya-Long Yu,1 Qian Tao,1 Chun-Mu Feng,1 and Guang-Han Cao1, 2, 3, †

1Department of Physics, Zhejiang University, Hangzhou 310027, China2State Key Lab of Silicon Materials, Zhejiang University, Hangzhou 310027, China

3Collaborative Innovation Centre of Advanced Microstructures, Nanjing 210093, China

(Dated: January 9, 2018)

The intrinsically hole-doped RbEuFe4As4 exhibits bulk superconductivity at Tsc = 36.5 K andferromagnetic ordering in the Eu sublattice at Tm = 15 K. Here we present a hole-compensationstudy by introducing extra itinerant electrons via a Ni substitution in the ferromagneticsuperconductor RbEuFe4As4 with Tsc > Tm. With the Ni doping, Tsc decreases rapidly, and the Eu-spin ferromagnetism and its Tm remain unchanged. Consequently, the system RbEu(Fe1−xNix)4As4transforms into a superconducting ferromagnet with Tm > Tsc for 0.07 ≤ x ≤ 0.08. Theoccurrence of superconducting ferromagnets is attributed to the decoupling between Eu2+ spins andsuperconducting Cooper pairs. The superconducting and magnetic phase diagram is established,which additionally includes a recovered yet suppressed spin-density-wave state.

PACS numbers: 74.70.Xa, 74.25.Ha, 74.62.Dh, 75.30.-m

INTRODUCTION

Superconductivity (SC) and ferromagnetism (FM) arebasically antagonistic and incompatible [1, 2], and onlyin a very few cases can they coexist simultaneously ina single material [1, 3]. The relative robustness of SCand FM can be reflected by the superconducting criti-cal temperature Tsc and the (ferro)magnetic transitiontemperature Tm. Materials with Tsc > Tm were earliercalled “ferromagnetic superconductors” (FSCs) [1], andthose with Tm > Tsc were then termed “superconductingferromagnets” (SFMs) [4–6]. Generally, a ferromagneticexchange field prevails over the intrinsic superconductingupper critical field H∗

c2 for Tm ≤ Tsc, hence SFMsare particularly rare. So far, examples of SFMs onlyinclude U-based germanides with spin-triplet SC [7]and ruthenocuprates with spin-singlet high-temperatureSC [8], the latter of which actually exhibit the coexistencewith a canted antiferromagnetism. Note that theclassification into FSCs and SFMs is meaningful forstudying the way of coexistence of the two antagonisticphenomena [5, 9].

In recent years, Eu-containing 122-type iron arsenideshave earned a lot of interest owing to the intriguinginterplay between SC and FM [3, 10]. The crystalstructure allows the magnetic Eu-atomic planes awayfrom the superconductively active Fe-atom sheets. Innon-doped EuFe2As2, the Eu sublattice is of an A-type antiferromagnetism below ∼19 K while the Fesublattice exhibits a spin-density wave (SDW) orderbelow ∼190 K [11–14]. SC at Tsc = 20-30 Kcan be induced by the chemical doping with P [15],Co [16], Ru [17], Ir [18, 19], etc. Simultaneously,the Eu2+ local spins become ferromagnetically orderedat Tm ∼ 18 K [20–24]. It has been concludedthat SC appears only when Tsc > Tm in systems of

EuFe2(As1−xPx)2 [25, 26], Eu(Fe1−xCox)2As2 [27], andSr1−yEuy(Fe0.88Co0.12)2As2 [28]. The conclusion also fitswith the absence of SC in Eu(Fe1−xNix)2As2 [29], sincethe Eu-free analogous system Sr(Fe1−xNix)2As2 shows amaximum Tsc of 9.8 K, which is significantly lower thanthe expected Tm [30].

Very recently, a variant of EuFe2As2, i.e. the 1144-typeAEuFe4As4 (A = Rb and Cs), were synthesized and char-acterized [31–33]. The twin compounds adopt a crystalstructure identical to that previously designed [34], whichwas first realized in AeAFe4As4 (Ae = Ca, Sr; A = K,Rb, Cs) [35]. In RbEuFe4As4, the Rb

+ and Eu2+ planes,sandwiched by FeAs layers, stack alternately along the caxis. The structure can also be viewed as an intergrowthbetween non-doped EuFe2As2 and heavily over-dopedRbFe2As2. As a result, RbEuFe4As4 is intrinsically hole-doped, exhibiting SC at Tsc = 36.5 K without any SDWordering [31, 32]. Additionally, evidence of FM of theEu2+ spins below Tm = 15 K is given by magnetizationmeasurements [32]. Compared with 122-type FSCs, theTsc value is significantly higher, and the Tm value isslightly lower. Important to be noted is that the Tsc

value of RbEuFe4As4 are almost the same as, or evenlarger than, those of the non-magnetic analogues (e.g.Tsc = 35.1 K in RbSrFe4As4 [35]), indicating that theEu2+ spins hardly break superconducting Cooper pairs.In this context, SC may survive easily in the presence ofEu-spin order, and therefore, it is of interest to seek foran SFM in the 1144-type system.

Now that RbEuFe4As4 bears an intrinsic hole doping(0.25 holes per Fe atom), it is natural to tune theTsc value by hole depletion via electron doping. Ourpreliminary trial with a Ba-for-Rb substitution wasdemonstrated unsuccessful, since a 122-type phase, in-stead of the expected 1144-type phase, became stabilized.Then, we turned to a substitution at the Fe site. To

2

compensate the doped holes more effectively, we chose Nias the dopant, because Ni2+ (3d8) has two more itinerantelectrons than Fe2+ (3d6) does, and more importantly,previous Ni-doping studies indeed show such an effect ofelectron doping [30, 36, 37].

In this paper, we report a systematic investigationon the magnetic and superconducting properties inRbEu(Fe1−xNix)4As4. As expected, Tsc decreasesrapidly with the Ni doping. On the other hand, theEu-spin FM and its Tm value remain unchanged. Thisleads to a discovery of SFMs in RbEu(Fe1−xNix)4As4with 0.07 ≤ x ≤ 0.08, in which SC survives when Tsc

becomes lower than Tm. The SFMs are found to showno Meissner state with a broadened resistive transition,because they are always under the internal field generatedby the FM of Eu2+ spins. The superconducting andmagnetic phase diagram has been established, and thereason for the existence of SFMs is discussed.

EXPERIMENTAL METHODS

Polycrystalline samples of RbEu(Fe1−xNix)4As4 with0 ≤ x ≤ 0.125 were prepared by solid-state reactionsin evacuated quartz ampoules sealed, similar to ourprevious report [32]. The source materials were theconstituent elements: Rb (99.75%), As (99.999%), Eu(99.9%), Fe (99.998%), and Ni (99.99%), all from AlfaAesar. Firstly, precursors of EuAs, FeAs, NiAs, andRbFe2As2 (with 5% excess of Rb) were prepared bysolid-state reactions in evacuated quartz tubes at 873-1023 K for 24 hours. These precursors and additional Fepowders were then mixed together in the nominal com-position of RbEu(Fe1−xNix)4As4, followed by thoroughlyhomogenizing with ball milling in an Ar-filled glove box.Secondly, the mixtures were pressed into pellets whichwere loaded in an alumina container jacketed with adouble-layer protector (a sealed Ta tube inside and aquartz ampoule outside). Finally, the samples wererapidly heated to 1133 K, holding for 20 hours, endedwith quenching in cool water. To improve samples’purity, the synthesis was repeated once or twice, withan intermediate grinding.

Powder x-ray diffraction (XRD) was performed on aPANalytical x-ray diffractometer with a monochromaticCu-Kα1 radiation at room temperature. The latticeparameters were obtained by a least-squares fit of 15-25 reflections in the range of 5◦ ≤ 2θ ≤ 80◦. Thesample’s chemical composition was checked by energy-dispersive x-ray spectroscopy (EDS). The resistivityand specific-heat measurements were conducted on aQuantum Design Physical Property Measurement Sys-tem (PPMS-9). The dc magnetization was measuredon a Quantum Design Magnetic Property MeasurementSystem (MPMS-XL5).

10 20 30 40 50 60

0.00 0.05 0.103.88

3.89

3.90

0.00 0.05 0.1013.24

13.26

13.28

13.30

*

2 (degree)

213

116

105 20

0

114

113

112

110

103

004

002

x = 0.1

x = 0.09

x = 0.08

x = 0.07

x = 0.06

x = 0.05

x = 0.025

x = 0

Inte

nsity

(arb

. uni

ts)

(a)

001

RbEu(Fe1-xNix)4As4

** FeAs

(b)

x

a (Å

)

(c)

x

c (Å

)

FIG. 1. (a) Powder X-ray diffraction patterns ofRbEu(Fe1−xNix)4As4 (0 ≤ x ≤ 0.1) at room temperature.(b) and (c) Lattice parameters a and c as functions of thenominal Ni concentration x. The dashed line in (c) gives thelinear fit.

RESULTS AND DISCUSSION

X-ray diffraction

Figure 1(a) shows XRD patterns of the series samplesof RbEu(Fe1−xNix)4As4 (0 ≤ x ≤ 0.1), which can bewell indexed by a tetragonal unit cell with a ≈ 3.89 Aand c ≈ 13.3 A. The variations in relative intensity ismainly due to (00l) preferred orientations. One sees thatall the samples from x = 0 to 0.1 are nearly single phase(only small amount of impurities such as FeAs appearin some of the samples). In the case of x = 0.125, theXRD pattern (not shown) indicates formation of 122-typephase. Therefore, the present study is limited to sampleswith 0 ≤ x ≤ 0.1.

The lattice constants were determined by a least-squares fit, the results of which are displayed in Table I.Figs. 1(b,c) plot the fitted lattice parameters a and c,respectively, as a function of the nominal Ni content.As is shown, while the a axis of the unit cell basicallyremains unchanged, the c axis decreases significantlywith the Ni doping. The result is quite similar tothose in Eu(Fe1−xNix)2As2 [29], LaFe1−xNixAsO [36],and Ba(Fe1−xNix)2As2 [37] systems. The linear decreasein c, which obeys the Vegard’s law, confirms that thedopant Ni indeed substitutes for Fe. The EDS on the

3

TABLE I. Room-temperature lattice constants and physical-property parameters of RbEu(Fe1−xNix)4As4 (0 ≤ x ≤ 0.1).The digits in parentheses give twice of the standard deviation of the least-squares fit. T ρ

sc is the superconducting midpointtransition temperature in the resistivity measurement. TSDW denotes the spin-density-wave transition temperature. Tm and Θare the magnetic-transition and Curie-Weiss temperatures, respectively. µeff and Msat are the effective magnetic moment inthe paramagnetic state and the ordered moment in the ferromagnetic state, respectively. Hcoe refers to the apparent coercivefield.

Lattice Constants Physical-Property Parametersx a(A) c (A) T ρ

sc (K) TSDW (K) Tm (K) Θ (K) µeff (µB/Eu) Msat (µB/Eu) Hcoe (Oe)0 3.8891(4) 13.304(17) 36.4 - 15.0 23.6 7.95 6.5 360

0.025 3.8900(6) 13.284(19) 30.3 - 15.0 24.3 7.88 6.5 2580.05 3.8909(3) 13.274(12) 23.0 28.9 15.0 24.5 8.00 6.4 880.06 3.8878(6) 13.268(27) 18.1 31.1 15.0 24.3 7.65 5.9 670.07 3.8889(7) 13.263(37) 11.2 35.0 15.1 24.2 7.46 5.9 440.08 3.8877(11) 13.261(48) 2.1 33.6 14.7 24.4 7.85 6.5 210.09 3.8892(8) 13.256(40) - 31.3 14.7 24.8 7.79 6.0 200.1 3.8877(4) 13.250(17) - 29.4 14.8 24.4 7.74 6.3 24

sample of x = 0.1 shows that the Ni content is 0.089(8),which remains unchanged throughout the sample. Thisconfirms that the sample is homogeneous for the Nidoping, and the actual Ni-doping level is close to thenominal one within the measurement uncertainty.

Magnetic properties

We first address the Eu-spin state by focus-ing on the high-temperature magnetic properties inRbEu(Fe1−xNix)4As4. As shown in Fig. 2, the magneticsusceptibility (χ) at high temperatures is of Curie-Weiss-type paramagnetism. The χ(T ) data in the temperaturerange of 50 K ≤ T ≤ 300 K can be well fitted with anextended Curie-Weiss law, χ = χ0 + C/(T − Θ), whereχ0 is the temperature-independent term, C gives theCurie constant from which the effective local moment isderived, and Θ represents the Curie-Weiss temperature.The derived effective moment µeff (see Table I) rangesbetween 7.46 and 8.00 µB/f.u. (f.u. refers to formulaunit), independent of the Ni substitution. These µeff

values are close to that (7.9 µB) expected for a free Eu2+

ion, indicating a spin state with total spins of S = 7/2 forthe Eu2+ ions. The Θ values fitted (from 23.6 to 24.8 K)is also independent of the Ni doping. The positive sign ofΘ reflects dominant ferromagnetic interactions betweenEu2+ spins.

At low temperatures, the system undergoes super-conducting and/or magnetic transitions. To determinetheir transition temperatures, Tsc and Tm, we performedmeasurement of temperature-dependent magnetization,M(T ), under a low field of 10 Oe. In general, asuperconducting transition can be easily recognized bythe strong diamagnetic signal owing to Meissner effect.However, in cases of the coexistence between SC and FMwith Tsc ≤ Tm, the diamagnetic signal may be coveredup by FM. For the ferromagnetic transition, on the other

0 50 100 150 200 250 3000.0

0.1

0.2

0.3

0.4

eff = 8.00 B/Eu = 24.5 K

RbEu(Fe1-xNix)4As4 (x = 0.05)H = 1 kOe

(em

u/m

ol)

T (K)

0

10

20

30

1/ (m

ol/e

mu)

FIG. 2. Temperature dependence of dc magnetic susceptibil-ity (χ = M/H) for a typical sample of RbEu(Fe1−xNix)4As4with x = 0.05 at a magnetic field of H = 1 kOe. Theright-hand axis plots 1/χ, indicating dominant Curie-Weissparamagnetism above 25 K. The data in the range of 50 K≤ T ≤ 300 K are fitted with Curie-Weiss law (displayed withsolid lines), from which the effective local moment µeff andthe Curie-Weiss temperature Θ are extracted as shown. Seethe text for details.

hand, the Curie temperature is traditionally determinedby Arrot approach [38]. However, the presence of SCmakes the method invalid. Here we take advantageof the magnetic hysteresis arising from the appearanceof magnetic domains. Namely, Tm is defined by thebifurcation temperature between field-cooling (FC) andzero-field-cooling (ZFC) M(T ) data measured undermagnetic fields lower than the coercive field. In fact, thebifurcation point basically coincides with the kink (peak)in the FC (ZFC) curves. Furthermore, the resultantTm value is precisely consistent with the heat-capacitymeasurement [32]. Note that a type-II SC may also giverise to a bifurcation between FC and ZFC curves owingto magnetic flux-pinning effect.

Figure 3(a-h) shows the low-field M(T ) data for

4

0 10 20 30 40

-0.2

-0.1

0.0

0 10 20 30 40-0.2

-0.1

0.0

0.1

0 10 20 30 40

0.0

0.4

0.8

0 10 20 30 40

0.0

0.4

0.8

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

0 10 20 30 40

0.0

0.2

0.4

0 10 20 30 40

0.0

0.2

0.4

0 10 20 30 40

0.0

0.2

0.4

0.6

21 22 23 24 15 16

15 16 14 15 16 14 15 16 14 15 16

Tsc

M (e

mu/

g)

T (K)

x = 0(a)

Tm

FC ZFC

x = 0.025

FC ZFC

TmTsc

(b)

M (e

mu/

g)

T (K)

x = 0.05 FC ZFC

Tm

Tsc

(c)

M (e

mu/

g)

T (K)

x = 0.06 FC ZFC Tm

(d)

M (e

mu/

g)

T (K)

FC ZFC

x = 0.07(e)

M (e

mu/

g)

T (K)

FC ZFC

x = 0.08(f)

M (e

mu/

g)

T (K)

FC ZFC

x = 0.09(g)

M (e

mu/

g)

T (K)

FC ZFC

x = 0.1(h)

M (e

mu/

g)

T (K)

Tsc

M

T (K)

TmTsc

M

T (K)

MFC

-MZF

C

0

Tm

M

T (K)

Tm

M

T (K)

Tm

M

T (K)

Tm

M

T (K)

FIG. 3. Temperature dependence of magnetization under a magnetic field of 10 Oe for RbEu(Fe1−xNix)4As4 (0 ≤ x ≤

0.1) samples. FC (solid symbols) and ZFC (open symbols) denote field cooling and zero-field cooling, respectively, in themagnetic measurements. Tsc and Tm marked with arrows represent the superconducting and magnetic transition temperatures,respectively. The insets show close-ups near the superconducting/magnetic transitions. The inset in panel (d) also plots thedifference between the FC and ZFC data, using the righ-hand axis.

samples of the RbEu(Fe1−xNix)4As4 series. The non-doped (x = 0) compound shows SC at Tsc = 36.5 Kand FM below Tm = 15 K [32]. For x = 0.025, thesuperconducting transition keeps robust, yet with anobviously reduced Tsc of 30.5 K. At x = 0.05, in whichTsc is further decreased to 22 K, the superconductingtransition becomes much less remarkable. For x = 0.06,only weak signature of SC at 15.7 K can be tracedfrom the slight difference between the FC and ZFC data,the latter of which comes from flux-pinning effect. Theexistence of SC is also evidenced from the diamagnetismin the ZFC data at lower temperatures and, in particular,from the zero resistance at 16 K in the resistivitymeasurement shown below. In the cases of x ≥ 0.07,no magnetic signal for SC can be detected, although theresistivity measurement clearly shows superconductingtransitions at lower temperatures for 0.07 ≤ x ≤ 0.08.

In contrast with the monotonic suppression in Tsc

with increasing Ni doping, the magnetic transitiontemperature Tm almost keeps unchanged. Note that theCurie-Weiss temperature Θ, which is remarkably higherthan Tm, does not depend on the Ni doping either.The lower-than-expected Tm value is probably related tothe quasi-two-dimensional magnetism caused by a muchweaker magnetic coupling along the c axis [32]. We willdiscuss on the magnetic interactions later on.

One may also note that, for lower Ni doping with x ≤

0.05, the FC magnetization decreases with decreasingtemperature just below Tm, showing a peak-like anomaly(PLA) at Tm, which casts doubt on the nature of the

0 10 20 300

2

4

6

0 10 20 300

2

4

6

0 10 20 300

2

4

6

0 10 20 300

2

4

6

0 10 20 300

2

4

6

0 10 20 300

2

4

6

x = 0.05 H = 0.1 kOe 0.5 kOe 1 kOe 2 kOe 5 kOe 10 kOe

M (

B/f.u

.)T (K)

(b)

(f) H = 0.1 kOe 0.5 kOe 1 kOe 2 kOe 5 kOe 10 kOe

M (

B/f.u

.)

T (K)

x = 0.1(e) x = 0.08 H = 0.1 kOe 0.5 kOe 1 kOe 2 kOe 5 kOe 10 kOe

M (

B/f.u

.)

T (K)

(a) x = 0.025 H = 0.1 kOe 0.5 kOe 1 kOe 2 kOe 5 kOe 10 kOe

M (

B/f.u

.)

T (K)

FC ZFC

H = 0.1 kOe 0.5 kOe 1 kOe 2 kOe

(c) x = 0.06

M (

B/f.u

.)

T (K)

H = 0.1 kOe 0.5 kOe 1 kOe 2 kOe

(d)x = 0.07

M (

B/f.u

.)

T (K)

FIG. 4. Temperature dependence of field-cooling (FC,solid symbols) and zero-field-cooling (ZFC, open symbols)magnetization under different magnetic fields for x = 0.025(a), 0.05 (b), 0.06 (c), 0.07 (d), 0.08 (e), and 0.1 (f) inRbEu(Fe1−xNix)4As4. The magnetization is conversed intoBohr magnetons per formula unit.

5

magnetic transition. To address this issue, we measuredthe M(T ) data at elevated magnetic fields shown inFig. 4. One sees that the PLA disappears under relativelylow fields (∼0.5 kOe). At higher fields with H ≥

5 kOe, all the M(T ) curves become almost identicalwithout bifurcations in the FC and ZFC data, suggestingcommonality of FM in the system. In fact, a similar PLAbehavior is also seen in 122-type FSCs [16, 39] wherean Eu-spin FM is unambiguously demonstrated [20–22].Note that the PLA here happens only for Tsc > Tm. Thissuggests that the PLA is probably in relation with thepresence of SC. As is pointed out theoretically, the stateof FM may be modified into crypto-ferromagnetism [40]or dense-domain structure [41] owing to the presence ofSC. The expected fine ferromagnetic domains that areantiferromagnetically aligned could give rise to a PLAin the FC M(T ) curve. Future studies with single-crystalline samples by magnetic-force microscopy [42]seem promising to clarify this issue.

The Eu-spin FM in RbEu(Fe1−xNix)4As4 is furtherconfirmed by the isothermal magnetization, M(H),shown in Fig. 5(a-f). At high temperatures (e.g., 40K), the M(H) data are essentially linear, consistent withthe paramagnetic state of Eu2+ spins. At temperaturesbelow Tm, by contrast, the M(H) curves are of an Sshape, characteristic of an FM. The saturation magne-tization Msat, defined here as the magnetization valueat 1 T and 2 K, scatters from 5.9 to 6.5 µB/f.u. (seeTable I). Samples containing more FeAs impurity tend tohave relatively low Msat (and µeff also). Other samplesshow an Msat value that is close to the expected one(7.0 µB per Eu2+) [43], indicating that Eu2+ spins orderferromagnetically.

Note that for samples of x ≤ 0.05, which have higherTsc values, the magnetic hysteresis is extended to highfields where the magnetization is saturated. This clearlyindicates the existence of type-II SC that commonlyexhibits flux-pinning effect, the latter of which also givesrise to an enhancement of the apparent coercive fieldHcoe. As shown in Table I, the intrinsic Hcoe, givenby non-superconducting samples of 0.08 < x ≤ 0.1, isactually around 20 Oe. The sample of x = 0.07 shows anenhanced Hcoe of 44 Oe, implying the existence of SC.Indeed, the resistivity measurement below demonstratesa superconducting transition at Tsc = 11 K, 4 K lowerthan Tm. That is to say, the sample is actually an SFM.Notably, no superconducting diamagnetism is detectedby the M(T ) data [Fig. 3(e)] and the virgin M(H)curve [Fig. 5(c)]. This observation is consistent withthe absence of Meissner state, as expected from theinternal field (∼4.5 kOe [44]) that is much higher than theintrinsic lower critical field. It is of great interest for thefuture to look into the anisotropic magnetic propertieswith using the single-crystalline samples.

-10 -5 0 5 10

-6

-3

0

3

6

-10 -5 0 5 10

-6

-3

0

3

6

-0.2 0.0 0.2-0.3

0.0

0.3

-0.1 0.0 0.1-0.2

0.0

0.2

-10 -5 0 5 10

-6

-3

0

3

6

-10 -5 0 5 10

-6

-3

0

3

6

-0.4 0.0 0.4-0.3

0.0

0.3

-0.1 0.0 0.1-0.2

0.0

0.2

-10 -5 0 5 10

-6

-3

0

3

6

-10 -5 0 5 10

-6

-3

0

3

6

-0.1 0.0 0.1-0.2

0.0

0.2

-0.1 0.0 0.1-0.2

0.0

0.2

2 K 10 K 20 K 40 K

x = 0.05

(b)

M (

B/f.u

.)

H (kOe)

2 K 10 K 20 K 40 K

x = 0.1

(f)

M (

B/f.u

.)

H (kOe)

H (kOe)

H (kOe)

2 K 10 K 20 K 40 K

M (

B/f.u

.)

H (kOe)

x = 0.025

(a)

2 K 10 K 20 K 40 K

x = 0.08

(e)

M (

B/f.u

.)H (kOe)

H (kOe)

H (kOe)

2 K 10 K 20 K

x = 0.06

(c)

M (

B/f.u

.)

H (kOe)

2 K 10 K 20 K

x = 0.07

(d)

M (

B/f.u

.)

H (kOe)

H (kOe)

H (kOe)

FIG. 5. Isothermal magnetization at low temperatures inthe RbEu(Fe1−xNix)4As4 series. The insets show close-upsof the magnetic hysteresis at 2 K, from which the virginmagnetization as well as the coercive field can be seen clearly.

Electrical resistivity

Figure 6 shows the electrical resistivity (ρ) forRbEu(Fe1−xNix)4As4 polycrystalline samples. To high-light the evolution of the temperature dependence, andalso to present the superconducting/SDW transitionsclearly, we normalize the ρ(T ) data relative to theresistivity values at 200 and 50 K, respectively. Firstof all, the slope dρ/dT in the normal state decreasesmonotonically with the Ni doping, giving rise to anincrease in the residual resistivity at low temperatures.This is consistent with the increase of Fe-site disorderwith Ni doping. Secondly, the superconducting transitiontemperature T ρ

sc decreases with the Ni doping, and SCis completely suppressed at x = 0.09 (the slight droparound 5 K is probably due to sample’s inhomogeneity).The result is basically consistent with the magneticmeasurement for x ≥ 0.06 with Tsc > Tm. In thecase of Tsc ≤ Tm, nevertheless, SC cannot be directlydetected in the magnetic measurement above, andthe resistive transitions become remarkably broadened.The broadened resistive transition is similar to theobservation in Eu(Fe0.81Co0.19)2As2 [45] also with Tsc <Tm, which can be explained in terms of the dissipativeflow of spontaneous vortices [9].

Another interestingly point is that the samples with

6

0 50 100 150 2000.0

0.2

0.4

0.6

0.8

0 10 20 30 40 500.7

0.8

0.9

1.0

x = 0.1RbEu(Fe1-xNix)4As4

x = 0 x = 0.025 x = 0.05 x = 0.06 x = 0.07 x = 0.075 x = 0.08 x = 0.09 x = 0.1

/

T (K)

(a)

x = 0

TSDW

(b)

x = 0.025 0.05 0.06 0.07 0.075 0.08 0.09 0.1

/

T (K)

Tm

FIG. 6. Temperature dependence of resistivity forRbEu(Fe1−xNix)4As4 polycrystalline samples at zero field.The top and bottom panels adopt a normalized scale relativeto the resistivity values at 200 and 50 K, respectively. Inthe bottom panel, the arrows labeled with TSDW point to theresistivity upturn that probably comes from a spin-densitywave (SDW) transition. The left dashed line labeled withTm marks the tiny kinks at which the Eu2+ spins orderferromagnetically.

x ≥ 0.05 show a resistivity upturn above Tsc, which isprobably due to a spin-density-wave (SDW) ordering inthe Fe sublattice. At x = 0.05, the hole concentration isreduced to nh = 0.15 holes per Fe atom, if assuming thatevery doped Ni atom depletes two holes. In the prototypeBa1−xKxFe2As2 system, SDW order remains at nh =0.15 [46]. Note that the SDW transition temperatureTSDW here is much lower than expected. Furthermore,TSDW decreases with x in the high doping regime. Theseresults can be ascribed to the Fe-site disorder mentionedabove. Similarly, a recovery of SDW by charge com-pensation was reported in Ba1−xKxFe1.86Co0.14As2 [47]and Eu0.5K0.5(Fe1−xNix)2As2 [48] systems. Here we notethat, according to a recent report on Ni- and Co-dopedCaKFe4As4 [49], the recovered SDW phase may havestrikingly different magnetic order.

In addition to the SDW-like anomaly above, Fig. 6(b)also shows a very slight (yet observable) kink at Tm. This

0 10 20 30 400

20

40

60

2 4 60.00

0.02

0.04

20 30 400.00

0.05

0.10

0.15

0 10 20 30 40 500

20

40

60

80

100

15 20 251.2

1.3

1.4

(b)

C (J

K-1 m

ol-1)

T (K)

B (x = 0.075) A (x = 0.1)

RbEu(Fe1-xNix)4As4

D/T

(J K

-2 m

ol-1)

T (K)

D = CB - CA

D = CB - CA

TSDW

D/T

(J K

-2 m

ol-1)

T (K)

*C

(J K

-1 m

ol-1)

T (K)

x = 0 x = 0.05 x = 0.06

RbEu(Fe1-xNix)4As4

(a)

C/T

(J K

-2 m

ol-1)

T (K)

FIG. 7. Temperature dependence of specific heat capacity (C)of the ferromagnetic superconductors (a) and the supercon-ducting ferromagnet (b) in RbEu(Fe1−xNix)4As4 under zeromagnetic field. The superconducting transitions are markedby arrows. Note that the insets in top and bottom panels plotC/T and D/T , respectively, where D stands for the specific-heat difference between samples of x = 0.075 and 0.1.

is due to the reduction of magnetic scattering on thecharge carriers when Eu2+ spins become ordered, akin tothe case in EuFe2As2 [11]. The tiny resistivity changeat Tm seems to reflect weak interactions between Eu2+

spins and the charge carriers.

Heat capacity

Figure 7 shows the temperature dependence of specificheat of RbEu(Fe1−xNix)4As4 under zero field, trackingfor superconducting transitions in the FSCs and SFMs.For x = 0, the specific-heat jump is clearly seen with∆C/Tsc = 208 mJ K−2 mol−1 at the superconductingtransition [32]. With increasing the Ni doping, thespecific-heat jump becomes unapparent. At x = 0.05and 0.06, for example, the ∆C/Tsc values are estimatedto be ∼50 and ∼20 mJ K−2 mol−1, respectively.Similar dramatic reduction in ∆C was also observedin underdoped Ba1−xKxFe2As2 [50]. As a comparison,

7

the underdoped Ba0.77K0.23Fe2As2 sample (Tsc = 23K) shows a ∆C/Tsc value of ∼40 mJ K−2 mol−1 [50],comparable to that of the x = 0.05 sample (with asimilar Tsc) in the present system. Note that thestrength of superconducting coupling varies with thedoping level [50], the former of which is reflected bythe dimensionless parameter, p = ∆C/γnTsc, where γndenotes the Sommerfeld coefficient in the normal state.The drastic reduction in ∆C for the underdoped samplesis actually a consequence of the concurrent decrease inall the three factors: γn, Tsc and p.In the case of SFMs with Tsc < Tm, no specific-

heat anomaly at Tsc can be directly seen from theraw data. Nevertheless, by subtraction of the specificheat between the superconducting (x = 0.075) andnon-superconducting (x = 0.1) samples, the specific-heat jump is still observable at 4.1 K (at which theresistivity drop to 6%), as shown in the upper-leftinset of Fig. 7(b) (note that the kink at ∼5 K isprobably due to an antiferromagnetic transition fromthe very small amount of Eu3O4 impurity [51]). Sincethe Sommerfeld coefficient is greatly reduced in theunderdoped regime [50], the ∆C/Tsc value (∼10 mJK−2 mol−1) is actually appreciable, which supports thebulk nature of superconductivity. Additionally, from thespecific-heat difference shown in the lower inset, one cansee another anomaly at ∼36 K (albeit no anomaly isobservable again in the raw data), in accordance withthe resistivity upturn. This anomaly, if being intrinsic,should be related to the recovered SDW transition.

Phase diagram

The results above allow us to construct the phasediagram in RbEu(Fe1−xNix)4As4, which is displayedin Fig. 8. As usual, the bottom axis employs thedirect control parameter, i.e. the Ni content x. Sincesubstitution of Fe2+ (3d6) with Ni2+ (3d8) doublycompensates the self-doped holes, the expected holeconcentration is nh = 0.25 − 2x, which is also shownin the middle horizontal axis. With the hole depletionby Ni doping, Tsc decreases monotonically. Note that Tsc

decreases more rapidly for x ≥ 0.05 where SDW orderappears, suggesting competing nature between SC andSDW. SC disappears at x > 0.08 or at nh ≤ 0.09. Onenotes that TSDW goes down for x ≥ 0.075.To understand the possible role of disorder,

we compare with the electronic phase diagram ofBa1−yKyFe2As2 [52] which is disorder-free at the Fe site.One sees that the Tsc values in RbEu(Fe1−xNix)4As4 areoverall lower in the doping area, and the critical holeconcentration (for appearance of SC) is significantlyhigher. On the other hand, the TSDW values are evenmuch lower, especially in the high doping regime. Bothresults strongly suggest that the disorder by Ni doping

FSC SFM

RbEu(Fe1-xNix)4As4

Tsc Tsc

TmEu TSDW

FSC

T (K

)

Ni Content, x

SC

FM

SDW

0.5 0.4 0.3 0.2 0.1

Ba1-yKyFe2As2 [44] Tsc

TSDW

K Content, y

0.20 0.15 0.10

Hole-Doping Level, nh

FIG. 8. Superconducting and magnetic phase diagram inRbEu(Fe1−xNix)4As4 containing various electronic phaseshighlighted with colors. T ρ

sc is the midpoint temperatureof superconducting resistive transitions, and the error barsdenote the transition widths. Tχ

sc is the onset temperatureof superconducting diamagnetic transitions. Other abbre-viations include: SC, superconductivity; SDW, spin-densitywave; FM, ferromagnet; FSC, ferromagnetic superconductor;SFM, superconducting ferromagnet. For comparison, thephase lines of Ba1−yKyFe2As2 [52] are plotted using the topaxis. Both horizontal axes share the same hole-doping level,nh = 0.25 − 2x = y/2, as shown in the middle axis.

plays an important role in suppressing SC as well asSDW.

Then, how much do the Eu2+ spins influence theTsc? As we emphasize in the Introduction, first of all,SC in RbEuFe4As4 is not suppressed at all. Secondly,the Tsc values in RbEu(Fe1−xNix)4As4 are on average∼10 K lower than those in Ba1−yKyFe2As2 [52]. Theamount of Tsc reduction is very close to that of Eu-free Ba1−yKyFe1.86Co0.14As2 (compared in the sameway with Ba1−yKyFe2As2) [47], suggesting that disorderplays the dominant role for suppression of Tsc. In otherwords, the Eu2+ spins play a relatively minor (if not noneat all) role in suppressing SC. Thirdly, the Tsc valuesof RbEu(Fe1−xNix)4As4 are even higher than those inEu-diluted Eu0.5K0.5(Fe1−xNix)2As2 [48], the latter ofwhich shows an antiferromagnetism (e.g., TN = 8.5 Kfor x = 0.08) for Eu2+ spins which implies relatively lessinfluence on SC. This comparison further corroboratesthat Eu2+ spins hardly suppress Tsc in the 1144-typesystem.

In contrast to the dramatic changes in the statesassociated with the Fe sublattice, the ferromagnetic statein the Eu sublattice remains unchanged. In particular,Tm does not depend on the Ni doping, resulting in a

8

crossing at x ∼ 0.065 between the data lines of Tsc andTm. According to the classification for materials withcoexistence of SC and FM [5, 9], the system changesfrom FSC to SFM at the crossing point. For x > 0.08,the system shows coexistence of Eu-spin FM and Fe-siteSDW below 15 K. Although the sample with x = 0.125(corresponding to nh = 0) could not be synthesized owingto the solubility limit, one may expect by extrapolationthat this completely hole-compensated material wouldshow a similar behavior to that of the x = 0.1 sample.

Discussion

In the following, we discuss why both FSCs andSFMs exist in RbEu(Fe1−xNix)4As4 system. First ofall, coexistence of SC and FM in Eu-containing 122-type iron pnictides was tentatively explained in termsof Fe-3d multi-orbitals and robustness of SC [3]. Onthe one hand, multi-3d-orbitals in the valence bandallow both SC mainly from 3dyz/zx electrons and Eu-spin FM via Ruderman-Kittel-Kasuya-Yosida (RKKY)interactions through 3dx2−y2 and 3dz2 electrons. On theother hand, the intrinsic upper critical field H∗

c2 of thesuperconductors alike (e.g., an Eu-free analogue) couldbe high enough to overcome the exchange field betweenEu2+ spins and Cooper pairs.In this context, the absence of SC for Tsc < Tm

in EuFe2(As1−xPx)2 [25, 26], Eu(Fe1−xNix)2As2 [29],and Eu(Fe1−xCox)2As2 [27] can be attributed to therelatively low H∗

c2 in relation with the lower Tsc. Herewe note that the Eu(Fe0.81Co0.19)2As2 single crystalsgrown from Sn flux were reported to show SC withTsc < Tm [45]. The possible reason is that the Eu-spin exchange field could be reduced due to existenceof defects in the Eu sublattice. One notes that theSn-flux-grown “Eu(Fe0.82Co0.18)2As2” samples showedSDW order [53], indicating that they were actually in anunderdoped regime. The underdoped status with heavyCo-doping levels suggests the possibility of significantEu deficiencies. Besides, the flux Sn could also beincorporated into the Eu site [54]. Both factors lead todilution in the Eu sublattice, such that the exchange fieldthat breaks Cooper pairs may be reduced, which helpsthe survival of SC.In the 1144-type system of RbEu(Fe1−xNix)4As4,

the Eu2+ spins hardly suppress SC. Therefore, notonly do FSCs exist, but also SFMs occur inRbEu(Fe1−xNix)4As4. Note that the internal fieldgenerated from the Eu-spin FM is about 4.5 kOe, beinghigh enough to induce spontaneous vortices, yet not highenough to destroy SC.Finally, we comment on the magnetic interactions

between Eu2+ spins in RbEu(Fe1−xNix)4As4. Theexperimental fact is that neither Tm nor Θ changewith the Ni doping. This result contrasts to the

change in Tm from 20 to 16 K, accompanying withan antiferromagnetic-to-ferromagnetic transition, only by3% Ni doping in Eu(Fe1−xNix)2As2 [29]. The sensitivityto Ni doping dictates an indirect RKKY interactionwhose strength is proportional to cos(2kFr)/r

3, wherekF is the Fermi vector and, r is the distance betweenEu2+ spins. Conversely, the invariance of Tm and Θagainst electron doping in RbEu(Fe1−xNix)4As4 suggeststhat the RKKY interaction may not be the dominantmagnetic exchange interaction. This reminds us of theferromagnetic europium chalcogenides, EuO (Tm = 69.2K) and EuS (Tm = 16.6 K) [55], where there are noitinerant electrons for an indirect RKKY interaction.So, the effective ferromagnetic couplings between Eu2+

spins in RbEu(Fe1−xNix)4As4 may be due to the so-called d−f [56] and/or As−Eu−As superexchange inter-actions. Such exchange interactions naturally explainthe decoupling between Eu-4f and Fe-3d orbitals, whichconversely sheds light on the mechanism of iron-basedsuperconductivity. Future theoretical analyses andcalculations may help to clarify this issue.

CONCLUSION

In summary, we have systematically studiedthe magnetic and superconducting properties inRbEu(Fe1−xNix)4As4 (0 ≤ x ≤ 0.1). With the Nidoping that introduces extra itinerant electrons, the self-doped holes are gradually compensated. Resultantly, thesuperconducting transition temperature Tsc decreasesrapidly, and superconductivity disappears at x ∼ 0.08.The hole depletion also brings the recovery of SDWorder for x ≥ 0.05. For the Eu sublattice, the Eu-spin ferromagnetism in RbEuFe4As4 remains, and itsCurie temperature keeps unchanged. This gives riseto unique SFMs showing absence of Meissner state.The realization of crossover from FSCs to SFMs makesRbEu(Fe1−xNix)4As4 a promising playground to lookinto the interplay between SC and FM for the future.

This work was supported by the National NaturalScience Foundation of China (No. 11474252) andNational Key Research and Development Program ofChina (Nos. 2016YFA0300202).

∗ These authors contribute equally to this work.† ghcao@zju.edu.cn

[1] L. N. Bulaevskii, A. I. Buzdin, M. L. Kulic, and S. V.Panjukov, Coexistence of superconductivity and mag-netism theoretical predictions and experimental results,Adv. Phys. 34, 175 (1985).

[2] A. I. Buzdin, Proximity effects in

9

superconductor-ferromagnet heterostructures,Rev. Mod. Phys. 77, 935 (2005).

[3] G.-H. Cao, W.-H. Jiao, Y.-K. Luo, Z. Ren,S. Jiang, and Z.-A. Xu, Coexistence ofsuperconductivity and ferromagnetism in iron pnictides,J. Phys.: Conf. Ser. 391, 012123 (2012).

[4] E. B. Sonin and I. Felner, Spontaneous vortexphase in a superconducting weak ferromagnet,Phys. Rev. B 57, 14000(R) (1998).

[5] B. Lorenz and C.-W. Chu, Superconducting ferromag-nets: Ferromagnetic domains in the superconductingstate, Nat. Mater. 4, 516 (2005).

[6] Note that the terminology “ferromagnetic superconduc-tor” is also occasionally employed in the literature relatedto the U-based germanides [7], which show Tm > Tsc.

[7] V. P. Mineev, Superconductivity in uranium ferromag-nets, Physcis-Uspekhi 60, 121 (2017).

[8] T. Nachtrab, C. Bernhard, C. Lin, D. Koelle,and R. Kleiner, The ruthenocuprates: naturalsuperconductor-ferromagnet multilayers,Comptes Rendus Physique 7, 68 (2006).

[9] W.-H. Jiao, Q. Tao, Z. Ren, Y. Liu, and G.-H. Cao, Evidence of spontaneous vortex groundstate in an iron-based ferromagnetic superconductor,npj Quantum Materials 2, 50 (2017).

[10] S. Zapf and M. Dressel, Europium-based iron pnictides: aunique laboratory for magnetism, superconductivity andstructural effects, Rep. Prog. Phys. 80, 016501 (2017).

[11] Z. Ren, Z. Zhu, S. Jiang, X. Xu, Q. Tao, C. Wang,C. Feng, G. Cao, and Z. Xu, Antiferromagnetictransition in EuFe2As2: A possible parent compound forsuperconductors, Phys. Rev. B 78, 052501 (2008).

[12] S. Jiang, Y. Luo, Z. Ren, Z. Zhu, C. Wang,X. Xu, Q. Tao, G. Cao, and Z. Xu, Meta-magnetic transition in EuFe2As2 single crystals,New J. Phys. 11, 025007 (2009).

[13] J. Herrero-Martın, V. Scagnoli, C. Mazzoli, Y. Su,R. Mittal, Y. Xiao, T. Brueckel, N. Kumar, S. K. Dhar,A. Thamizhavel, and L. Paolasini, Magnetic structureof EuFe2As2 as determined by resonant x-ray scattering,Phys. Rev. B 80, 134411 (2009).

[14] Y. Xiao, Y. Su, M. Meven, R. Mittal, C. M. N. Kumar,T. Chatterji, S. Price, J. Persson, N. Kumar, S. K. Dhar,A. Thamizhavel, and T. Brueckel, Magnetic structure ofEuFe2As2 determined by single-crystal neutron diffrac-tion, Phys. Rev. B 80, 174424 (2009).

[15] Z. Ren, Q. Tao, S. Jiang, C. Feng, C. Wang,J. Dai, G. Cao, and Z. Xu, Superconductiv-ity induced by phosphorus doping and its coex-istence with ferromagnetism in EuFe2(As0.7P0.3)2,Phys. Rev. Lett. 102, 137002 (2009).

[16] S. Jiang, H. Xing, G. Xuan, Z. Ren, C. Wang,Z. Xu, and G. Cao, Superconductivity andlocal-moment magnetism in Eu(Fe0.89Co0.11)2As2,Phys. Rev. B 80, 184514 (2009).

[17] W.-H. Jiao, Q. Tao, J.-K. Bao, Y.-L. Sun, C.-M. Feng, Z.-A. Xu, I. Nowik, I. Felner, andG.-H. Cao, Anisotropic superconductivity inEu(Fe0.75Ru0.25)2As2 ferromagnetic superconductor,EPL (Europhysics Letters) 95, 67007 (2011).

[18] W.-H. Jiao, H.-F. Zhai, J.-K. Bao, Y.-K. Luo,Q. Tao, C.-M. Feng, Z.-A. Xu, and G.-H.Cao, Anomalous critical fields and the absenceof Meissner state in Eu(Fe0.88Ir0.12)2As2 crystals,

New J. Phys. 15, 113002 (2013).[19] U. B. Paramanik, D. Das, R. Prasad, and Z. Hos-

sain, Reentrant superconductivity in Eu(Fe1−xIrx)2As2,J. Phys.: Condens. Matter 25, 265701 (2013).

[20] S. Nandi, W. T. Jin, Y. Xiao, Y. Su, S. Price,D. K. Shukla, J. Strempfer, H. S. Jeevan, P. Gegen-wart, and T. Bruckel, Coexistence of supercon-ductivity and ferromagnetism in P-doped EuFe2As2,Phys. Rev. B 89, 014512 (2014).

[21] S. Nandi, W. T. Jin, Y. Xiao, Y. Su, S. Price,W. Schmidt, K. Schmalzl, T. Chatterji, H. S. Jee-van, P. Gegenwart, and T. Bruckel, Coexistence offerromagnetism and superconductivity in iron basedpnictides: a time resolved magnetooptical study,Phys. Rev. B 90, 094407 (2014).

[22] W. T. Jin, S. Nandi, Y. Xiao, Y. Su, O. Zaharko,Z. Guguchia, Z. Bukowski, S. Price, W. H.Jiao, G. H. Cao, and T. Bruckel, Magneticstructure of superconducting Eu(Fe0.82Co0.18)2As2as revealed by single-crystal neutron diffraction,Phys. Rev. B 88, 214516 (2013).

[23] W. T. Jin, W. Li, Y. Su, S. Nandi, Y. Xiao, W. H.Jiao, M. Meven, A. P. Sazonov, E. Feng, Y. Chen, C. S.Ting, G. H. Cao, and T. Bruckel, Magnetic ground stateof superconducting Eu(Fe0.88Ir0.12)2As2: A combinedneutron diffraction and first-principles calculation study,Phys. Rev. B 91, 064506 (2015).

[24] V. K. Anand, D. T. Adroja, A. Bhattacharyya, U. B.Paramanik, P. Manuel, A. D. Hillier, D. Khalyavin,and Z. Hossain, µSR and neutron diffraction investi-gations on the reentrant ferromagnetic superconductorEu(Fe0.86Ir0.14)2As2, Phys. Rev. B 91, 094427 (2015).

[25] G. Cao, S. Xu, Z. Ren, S. Jiang,C. Feng, and Z. Xu, Superconductivityand ferromagnetism in EuFe2(As1−xPx)2,J. Phys.: Condens. Matter 23, 464204 (2011).

[26] H. S. Jeevan, D. Kasinathan, H. Rosner, and P. Gegen-wart, Interplay of antiferromagnetism, ferromagnetism,and superconductivity in EuFe2(As1−xPx)2 single crys-tals, Phys. Rev. B 83, 054511 (2011).

[27] M. Nicklas, M. Kumar, E. Lengyel, W. Schnelle,and A. Leithe-Jasper, Competition of local-momentferromagnetism and superconductivity in Co-substitutedEuFe2As2, J. Phys.: Conf. Ser. 273, 012101 (2011).

[28] R. Hu, S. L. Bud’ko, W. E. Straszheim, andP. C. Canfield, Phase diagram of superconductivityand antiferromagnetism in single crystals ofSr(Fe1−xCox)2As2 and Sr1−yEuy(Fe0.88Co0.12)2As2,Phys. Rev. B 83, 094520 (2011).

[29] Z. Ren, X. Lin, Q. Tao, S. Jiang, Z. Zhu,C. Wang, G. Cao, and Z. Xu, Suppression of spin-density-wave transition and emergence of ferromag-netic ordering of Eu2+ moments in EuFe2−xNixAs2,Phys. Rev. B 79, 094426 (2009).

[30] S. R. Saha, N. P. Butch, K. Kirshenbaum,and J. Paglione, Evolution of bulksuperconductivity in SrFe2As2 with Ni substitution,Phys. Rev. B 79, 224519 (2009).

[31] K. Kawashima, T. Kinjo, T. Nishio, S. Ishida, H. Fu-jihisa, Y. Gotoh, K. Kihou, H. Eisaki, Y. Yoshida,and A. Iyo, Superconductivity in Fe-based compoundEuAFe4As4 (A = Rb and Cs), J. Phys. Soc. Jpn. 85,064710 (2016).

[32] Y. Liu, Y.-B. Liu, Z.-T. Tang, H. Jiang, Z.-C.

10

Wang, A. Ablimit, W.-H. Jiao, Q. Tao, C.-M.Feng, Z.-A. Xu, and G.-H. Cao, Superconductiv-ity and ferromagnetism in hole-doped RbEuFe4As4,Phys. Rev. B 93, 214503 (2016).

[33] Y. Liu, Y.-B. Liu, Q. Chen, Z.-T. Tang, W.-H. Jiao, Q. Tao, Z.-A. Xu, and G.-H. Cao,A new ferromagnetic superconductor: CsEuFe4As4,Sci. Bull. 61, 1213 (2016).

[34] H. Jiang, Y.-L. Sun, Z.-A. Xu, and G.-H. Cao,Crystal chemistry and structural design of iron-basedsuperconductors, Chin. Phys. B 22, 087410 (2013).

[35] A. Iyo, K. Kawashima, T. Kinjo, T. Nishio, S. Ishida,H. Fujihisa, Y. Gotoh, K. Kihou, H. Eisaki, andY. Yoshida, New-structure-type Fe-based superconduc-tors: CaAFe4As4 (A = K, Rb, Cs) and SrAFe4As4 (A =Rb, Cs), J. Am. Chem. Soc. 138, 3410 (2016).

[36] G. Cao, S. Jiang, X. Lin, C. Wang, Y. Li, Z. Ren,Q. Tao, C. Feng, J. Dai, Z. Xu, and F.-C. Zhang,Narrow superconducting window in LaFe1−xNixAsO,Phys. Rev. B 79, 174505 (2009).

[37] L. J. Li, Y. K. Luo, Q. B. Wang, H. Chen, Z. Ren, Q. Tao,Y. K. Li, X. Lin, M. He, Z. W. Zhu, G. H. Cao, and Z. A.Xu, Superconductivity induced by Ni doping in BaFe2As2single crystals, New J. Phys. 11, 025008 (2009).

[38] A. Arrott, Criterion for ferromagnetismfrom observations of magnetic isotherms,Phys. Rev. 108, 1394 (1957).

[39] S. Zapf, H. S. Jeevan, T. Ivek, F. Pfister, F. Klingert,S. Jiang, D. Wu, P. Gegenwart, R. K. Kremer, andM. Dressel, EuFe2(As1−xPx)2: Reentrant spin glass andsuperconductivity, Phys. Rev. Lett. 110, 237002 (2013).

[40] P. W. Anderson and H. Suhl, Spin alignment in thesuperconducting state, Phys. Rev. 116, 898 (1959).

[41] M. Faure and A. I. Buzdin, Domainstructure in a superconducting ferromagnet,Phys. Rev. Lett. 94, 187202 (2005).

[42] I. S. Veshchunov, L. Y. Vinnikov, V. S. Stolyarov,N. Zhou, Z. X. Shi, X. F. Xu, S. Y. Grebenchuk, D. S.Baranov, I. A. Golovchanskiy, S. Pyon, Y. Sun, W. Jiao,G. Cao, T. Tamegai, and A. A. Golubov, Visualizationof the magnetic flux structure in phosphorus-dopedEuFe2As2 single crystals, JETP Letters 105, 98 (2017).

[43] Note that the experimental value of Msat increasesfurther at lower temperatures and at higher magneticfields.

[44] The internal magnetic field generated by Eu-spin ferro-magnetism in RbEuFe4As4 is about a half of that (∼9kOe) for an Eu-based 122-type compound [9, 18], sincethe volume concentration of Eu moments of the formeris only a half of that of the latter.

[45] V. H. Tran, T. A. Zaleski, Z. Bukowski, L. M.Tran, and A. J. Zaleski, Tuning superconduc-tivity in Eu(Fe0.81Co0.19)2As2 with magnetic fields,

Phys. Rev. B 85, 052502 (2012).[46] H. Chen, Y. Ren, Y. Qiu, W. Bao, R. H.

Liu, G. Wu, T. Wu, Y. L. Xie, X. F. Wang,Q. Huang, and X. H. Chen, Coexistence of the spin-density wave and superconductivity in Ba1−xKxFe2As2,EPL (Europhysics Letters) 85, 17006 (2009).

[47] V. Zinth, T. Dellmann, H.-H. Klauss,and D. Johrendt, Recovery of aparentlike state in Ba1−xKxFe1.86Co0.14As2,Angew. Chem. Inter. Ed. 50, 7919 (2011).

[48] Anupam, V. K. Anand, P. L. Paulose, S. Ramakrishnan,

C. Geibel, and Z. Hossain, Effect of Ni doping onmagnetism and superconductivity in Eu0.5K0.5Fe2As2,Phys. Rev. B 85 (2012).

[49] W. R. Meier, Q.-P. Ding, A. Kreyssig, S. L. Bud’ko,A. Sapkota, K. Kothapalli, V. Borisov, R. Valenti, C. D.Batista, P. P. Orth, R. M. Fernandes, A. I. Goldman,Y. Furukawa, A. E. Bohmer, and P. C. Canfield,Hedgehog spin vortex crystal in a hole-doped iron basedsuperconductor, arXiv 1706, 01067 (2017).

[50] F. Hardy, A. E. Bohmer, L. de’ Medici, M. Capone,G. Giovannetti, R. Eder, L. Wang, M. He, T. Wolf,P. Schweiss, R. Heid, A. Herbig, P. Adelmann,R. A. Fisher, and C. Meingast, Strong cor-relations, strong coupling, and s-wave supercon-ductivity in hole-doped BaFe2As2 single crystals,Phys. Rev. B 94, 205113 (2016).

[51] L. Holmes and M. Schieber, MagneticOrdering in Eu3O4 and EuGd2O4,Journal of Applied Physics 37, 968 (1966).

[52] S. Avci, O. Chmaissem, D. Y. Chung, S. Rosenkranz,E. A. Goremychkin, J. P. Castellan, I. S. Todorov, J. A.Schlueter, H. Claus, A. Daoud-Aladine, D. D. Khalyavin,M. G. Kanatzidis, and R. Osborn, Phase diagram ofBa1−xKxFe2As2, Phys. Rev. B 85, 184507 (2012).

[53] W. T. Jin, Y. Xiao, Z. Bukowski, Y. Su, S. Nandi,A. P. Sazonov, M. Meven, O. Zaharko, S. Demirdis,K. Nemkovski, K. Schmalzl, L. M. Tran, Z. Guguchia,E. Feng, Z. Fu, and T. Bruckel, Phase diagram of Eumagnetic ordering in Sn-flux-grown Eu(Fe1−xCox)2As2single crystals, Phys. Rev. B 94, 184513 (2016).

[54] Y. Su, P. Link, A. Schneidewind, T. Wolf, P. Adel-mann, Y. Xiao, M. Meven, R. Mittal, M. Rotter,D. Johrendt, T. Brueckel, and M. Loewenhaupt, An-tiferromagnetic ordering and structural phase transitionin Ba2Fe2As2 with Sn incorporated from the growth flux,Phys. Rev. B 79, 064504 (2009).

[55] L. Passell, O. W. Dietrich, and J. Als-Nielsen,Neutron scattering from the Heisenberg ferromag-nets EuO and EuS. I. The exchange interactions,Phys. Rev. B 14, 4897 (1976).

[56] K. T., Exchange mechanisms in Europium chalcogenides,IBM J. Res. Dev. 14, 214 (1970).